Longitudinal wave

Longitudinal waves are waves which oscillate in

the direction which is parallel to the direction in
which the wave travels and displacement of the
medium is in the same (or opposite) direction of
the wave propagation. Mechanical longitudinal
waves are also called compressional or
compression waves, because they produce
compression and rarefaction when travelling
through a medium, and pressure waves, because
they produce increases and decreases in pressure.
A wave along the length of a stretched Slinky toy,
where the distance between coils increases and
A type of longitudinal wave: A plane pressure pulse
decreases, is a good visualization. Real-world wave.
examples include sound waves (vibrations in
pressure, a particle of displacement, and particle
velocity propagated in an elastic medium) and Nonfree image: detailed animation of
seismic P waves (created by earthquakes and a longitudinal wave
explosions). Detailed animation of longitudinal wave
motion (CC-BY-NC-ND 4.0) (http://www.acs.ps
The other main type of wave is the transverse wave,
u.edu/drussell/Demos/waves/Lwave-v8.gif)
in which the displacements of the medium are at
right angles to the direction of propagation.
Transverse waves, for instance, describe some bulk sound waves in solid materials (but not in
fluids); these are also called "shear waves" to differentiate them from the (longitudinal) pressure
waves that these materials also support.

Nomenclature
"Longitudinal waves" and "transverse waves" have been abbreviated by some authors as "L-waves"
and "T-waves", respectively, for their own convenience.[1] While these two abbreviations have
specific meanings in seismology (L-wave for Love wave[2] or long wave[3]) and electrocardiography
(see T wave), some authors chose to use "ℓ-waves" (lowercase 'L') and "t-waves" instead, although
they are not commonly found in physics writings except for some popular science books.[4]

Sound waves
For longitudinal harmonic sound waves, the frequency and wavelength can be described by the
formula

where:
 is the displacement of the point on the traveling sound wave;
is the distance from the point to the
wave's source;
is the time elapsed;
is the amplitude of the oscillations,
is the speed of the wave; and
is the angular frequency of the wave.

The quantity is the time that the wave takes to

travel the distance

The ordinary frequency ( ) of the wave is given by

Representation of the propagation of an

omnidirectional pulse wave on a 2‑D grid
The wavelength can be calculated as the relation (empirical shape)

between a wave's speed and ordinary frequency.

For sound waves, the amplitude of the wave is the difference between the pressure of the
undisturbed air and the maximum pressure caused by the wave.

Sound's propagation speed depends on the type, temperature, and composition of the medium
through which it propagates.

Speed of longitudinal waves

Isotropic medium
For isotropic solids and liquids, the speed of a longitudinal wave can be described by

where

is the elastic modulus, such that

where is the shear modulus and is the bulk modulus;
is the mass density of the medium.

Attenuation of longitudinal waves

The attenuation of a wave in a medium describes the loss of energy a wave carries as it propagates
throughout the medium.[5] This is caused by the scattering of the wave at interfaces, the loss of
energy due to the friction between molecules, or geometric divergence.[5] The study of attenuation
of elastic waves in materials has increased in recent years, particularly within the study of
polycrystalline materials where researchers aim to "nondestructively evaluate the degree of
damage of engineering components" and to "develop improved procedures for characterizing
microstructures" according to a research team led by R. Bruce Thompson in a Wave Motion
publication.[6]

Attenuation in viscoelastic materials

In viscoelastic materials, the attenuation coefficients per length for longitudinal waves and
for transverse waves must satisfy the following ratio:

where and are the transverse and longitudinal wave speeds respectively.[7]

Attenuation in polycrystalline materials

Polycrystalline materials are made up of various crystal grains which form the bulk material. Due
to the difference in crystal structure and properties of these grains, when a wave propagating
through a poly-crystal crosses a grain boundary, a scattering event occurs causing scattering based
attenuation of the wave.[8] Additionally it has been shown that the ratio rule for viscoelastic
materials,

applies equally successfully to polycrystalline materials.[8]

A current prediction for modeling attenuation of waves in polycrystalline materials with elongated
grains is the second-order approximation (SOA) model which accounts the second order of
inhomogeneity allowing for the consideration multiple scattering in the crystal system.[9][10] This
model predicts that the shape of the grains in a poly-crystal has little effect on attenuation.[9]

Pressure waves
The equations for sound in a fluid given above also apply to acoustic waves in an elastic solid.
Although solids also support transverse waves (known as S-waves in seismology), longitudinal
sound waves in the solid exist with a velocity and wave impedance dependent on the material's
density and its rigidity, the latter of which is described (as with sound in a gas) by the material's
bulk modulus.[11]

In May 2022, NASA reported the sonification (converting astronomical data associated with
pressure waves into sound) of the black hole at the center of the Perseus galaxy cluster.[12][13]
Electromagnetics
Maxwell's equations lead to the prediction of electromagnetic waves in a vacuum, which are strictly
transverse waves; due to the fact that they would need particles to vibrate upon, the electric and
magnetic fields of which the wave consists are perpendicular to the direction of the wave's
propagation.[14] However plasma waves are longitudinal since these are not electromagnetic waves
but density waves of charged particles, but which can couple to the electromagnetic field.[14][15][16]

After Heaviside's attempts to generalize Maxwell's equations, Heaviside concluded that

electromagnetic waves were not to be found as longitudinal waves in "free space" or homogeneous
media.[17] Maxwell's equations, as we now understand them, retain that conclusion: in free-space
or other uniform isotropic dielectrics, electro-magnetic waves are strictly transverse. However
electromagnetic waves can display a longitudinal component in the electric and/or magnetic fields
when traversing birefringent materials, or inhomogeneous materials especially at interfaces
(surface waves for instance) such as Zenneck waves.[18]

In the development of modern physics, Alexandru Proca (1897–1955) was known for developing
relativistic quantum field equations bearing his name (Proca's equations) which apply to the
massive vector spin-1 mesons. In recent decades some other theorists, such as Jean-Pierre Vigier
and Bo Lehnert of the Swedish Royal Society, have used the Proca equation in an attempt to
demonstrate photon mass[19] as a longitudinal electromagnetic component of Maxwell's equations,
suggesting that longitudinal electromagnetic waves could exist in a Dirac polarized vacuum.
However photon rest mass is strongly doubted by almost all physicists and is incompatible with the
Standard Model of physics.

See also
Transverse wave
Sound
Acoustic wave
P-wave
Plasma waves

References
1. Winkler, Erhard (1997). Stone in Architecture: Properties, durability (https://books.google.com/b
ooks?id=u9zt12_gE-AC). Springer Science & Business Media. pp. 55 (https://books.google.co
m/books?id=u9zt12_gE-AC&pg=PA55), 57 (https://books.google.com/books?id=u9zt12_gE-AC
&pg=PA57) – via Google books.
2. Allaby, M. (2008). A Dictionary of Earth Sciences (http://www.oxfordreference.com/oso/viewentr
y/10.1093$002facref$002f9780199211944.001.0001$002facref-9780199211944-e-4890;jsessi
onid=ECBC0E5982D11489C3ACF1C7F4D391F9) (3rd ed.). Oxford University Press – via
oxfordreference.com.
3. Stahl, Dean A.; Landen, Karen (2001). Abbreviations Dictionary (https://books.google.com/boo
ks?id=t3fLBQAAQBAJ&pg=PA618) (10th ed.). CRC Press. p. 618 – via Google books.
4. Milford, Francine (2016). The Tuning Fork (https://books.google.com/books?id=SK3QDQAAQB
AJ&pg=PA43). pp. 43–44.
 5. "Attenuation" (https://wiki.seg.org/wiki/Attenuation#:~:text=Attenuation%20%E2%80%94%20th
e%20falloff%20of%20a,which%20is%20the%20conversion%20of). SEG Wiki.
6. Thompson, R. Bruce; Margetan, F.J.; Haldipur, P.; Yu, L.; Li, A.; Panetta, P.; Wasan, H. (April
2008). "Scattering of elastic waves in simple and complex polycrystals" (https://doi.org/10.101
6/j.wavemoti.2007.09.008). Wave Motion. 45 (5): 655–674. Bibcode:2008WaMot..45..655T (htt
ps://ui.adsabs.harvard.edu/abs/2008WaMot..45..655T). doi:10.1016/j.wavemoti.2007.09.008 (h
ttps://doi.org/10.1016%2Fj.wavemoti.2007.09.008). ISSN 0165-2125 (https://search.worldcat.or
g/issn/0165-2125).
7. Norris, Andrew N. (2017). "An inequality for longitudinal and transverse wave attenuation
coefficients" (https://pubs.aip.org/jasa/article/141/1/475/1058243/An-inequality-for-longitudinal-
and-transverse-wave). The Journal of the Acoustical Society of America. 141 (1): 475–479.
arXiv:1605.04326 (https://arxiv.org/abs/1605.04326). Bibcode:2017ASAJ..141..475N (https://ui.
adsabs.harvard.edu/abs/2017ASAJ..141..475N). doi:10.1121/1.4974152 (https://doi.org/10.112
1%2F1.4974152). ISSN 0001-4966 (https://search.worldcat.org/issn/0001-4966).
PMID 28147617 (https://pubmed.ncbi.nlm.nih.gov/28147617) – via pubs.aip.org/jasa.
8. Kube, Christopher M.; Norris, Andrew N. (2017-04-01). "Bounds on the longitudinal and shear
wave attenuation ratio of polycrystalline materials" (https://pubs.aip.org/jasa/article/141/4/2633/
1059148/Bounds-on-the-longitudinal-and-shear-wave). The Journal of the Acoustical Society of
America. 141 (4): 2633–2636. Bibcode:2017ASAJ..141.2633K (https://ui.adsabs.harvard.edu/a
bs/2017ASAJ..141.2633K). doi:10.1121/1.4979980 (https://doi.org/10.1121%2F1.4979980).
ISSN 0001-4966 (https://search.worldcat.org/issn/0001-4966). PMID 28464650 (https://pubme
d.ncbi.nlm.nih.gov/28464650).
9. Huang, M.; Sha, G.; Huthwaite, P.; Rokhlin, S. I.; Lowe, M. J. S. (2021-04-01). "Longitudinal
wave attenuation in polycrystals with elongated grains: 3D numerical and analytical modeling"
(https://doi.org/10.1121%2F10.0003955). The Journal of the Acoustical Society of America.
149 (4): 2377–2394. Bibcode:2021ASAJ..149.2377H (https://ui.adsabs.harvard.edu/abs/2021A
SAJ..149.2377H). doi:10.1121/10.0003955 (https://doi.org/10.1121%2F10.0003955).
ISSN 0001-4966 (https://search.worldcat.org/issn/0001-4966). PMID 33940885 (https://pubme
d.ncbi.nlm.nih.gov/33940885).
10. Huang, M.; Sha, G.; Huthwaite, P.; Rokhlin, S. I.; Lowe, M. J. S. (2020-12-01). "Elastic wave
velocity dispersion in polycrystals with elongated grains: Theoretical and numerical analysis" (h
ttps://pubs.aip.org/jasa/article/148/6/3645/1056424/Elastic-wave-velocity-dispersion-in-polycrys
tals). The Journal of the Acoustical Society of America. 148 (6): 3645–3662.
Bibcode:2020ASAJ..148.3645H (https://ui.adsabs.harvard.edu/abs/2020ASAJ..148.3645H).
doi:10.1121/10.0002916 (https://doi.org/10.1121%2F10.0002916). hdl:10044/1/85906 (https://h
dl.handle.net/10044%2F1%2F85906). ISSN 0001-4966 (https://search.worldcat.org/issn/0001-
4966). PMID 33379920 (https://pubmed.ncbi.nlm.nih.gov/33379920).
11. Weisstein, Eric W., "P-Wave (http://scienceworld.wolfram.com/physics/P-Wave.html)". Eric
Weisstein's World of Science.
12. Watzke, Megan; Porter, Molly; Mohon, Lee (4 May 2022). "New NASA Black Hole Sonifications
with a Remix" (https://www.nasa.gov/mission_pages/chandra/news/new-nasa-black-hole-sonifi
cations-with-a-remix.html). NASA. Retrieved 11 May 2022.
13. Overbye, Dennis (7 May 2022). "Hear the Weird Sounds of a Black Hole Singing – As part of
an effort to "sonify" the cosmos, researchers have converted the pressure waves from a black
hole into an audible … something" (https://www.nytimes.com/2022/05/07/science/space/astron
omy-black-hole-sound.html). The New York Times. Retrieved 11 May 2022.
14. David J. Griffiths, Introduction to Electrodynamics, ISBN 0-13-805326-X
15. John D. Jackson, Classical Electrodynamics, ISBN 0-471-30932-X.
16. Gerald E. Marsh (1996), Force-free Magnetic Fields, World Scientific, ISBN 981-02-2497-4
17. Heaviside, Oliver, "Electromagnetic theory". Appendices: D. On compressional electric or
magnetic waves. Chelsea Pub Co; 3rd edition (1971) 082840237X
18. Corum, K. L., and J. F. Corum, "The Zenneck surface wave", Nikola Tesla, Lightning
Observations, and stationary waves, Appendix II. 1994.
19. Lakes, Roderic (1998). "Experimental Limits on the Photon Mass and Cosmic Magnetic Vector
Potential". Physical Review Letters. 80 (9): 1826–1829. Bibcode:1998PhRvL..80.1826L (http
s://ui.adsabs.harvard.edu/abs/1998PhRvL..80.1826L). doi:10.1103/PhysRevLett.80.1826 (http
s://doi.org/10.1103%2FPhysRevLett.80.1826).

Further reading
Varadan, V. K., and Vasundara V. Varadan, "Elastic wave scattering and propagation".
Attenuation due to scattering of ultrasonic compressional waves in granular media – A.J.
Devaney, H. Levine, and T. Plona. Ann Arbor, Mich., Ann Arbor Science, 1982.
Schaaf, John van der, Jaap C. Schouten, and Cor M. van den Bleek, "Experimental
Observation of Pressure Waves in Gas-Solids Fluidized Beds". American Institute of Chemical
Engineers. New York, N.Y., 1997.
Krishan, S.; Selim, A. A. (1968). "Generation of transverse waves by non-linear wave-wave
interaction". Plasma Physics. 10 (10): 931–937. Bibcode:1968PlPh...10..931K (https://ui.adsab
s.harvard.edu/abs/1968PlPh...10..931K). doi:10.1088/0032-1028/10/10/305 (https://doi.org/10.
1088%2F0032-1028%2F10%2F10%2F305).
Barrow, W.L. (1936). "Transmission of Electromagnetic Waves in Hollow Tubes of Metal".
Proceedings of the IRE. 24 (10): 1298–1328. doi:10.1109/JRPROC.1936.227357 (https://doi.or
g/10.1109%2FJRPROC.1936.227357). S2CID 32056359 (https://api.semanticscholar.org/Corp
usID:32056359).
Russell, Dan, "Longitudinal and Transverse Wave Motion". Acoustics Animations, Pennsylvania
State University, Graduate Program in Acoustics.
Longitudinal Waves, with animations "The Physics Classroom"

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